Watch London from the sofa, then go and see it late

The National Theatre, the Royal Ballet and Opera and Wigmore Hall all put their stages online, so you can watch London before you land or from the apartment sofa after a day of meetings. For the evenings when you want the real thing, a few big museums stay open well after the offices empty.
Theatre, opera and chamber music online
National Theatre at Home is the NT's subscription library of productions filmed on its South Bank stages, with apps for phones, tablets and smart TVs. It's the easiest way to catch a show you missed, and a decent way to decide whether something currently on is worth a West End ticket. For something closer to being there, National Theatre Live broadcasts selected productions to cinemas, and the Barbican runs a regular series of NT Live and encore screenings, handy on a weeknight when the play itself is sold out.
The Royal Opera House now goes by the Royal Ballet and Opera, and its service, Royal Ballet and Opera Stream, carries full-length opera and ballet recorded at Covent Garden, with a TV app as well as mobile and web versions. Watch a production there first and you'll know which seats in the house are worth paying for when you do go.
Wigmore Hall, the chamber music hall on Wigmore Street in Marylebone, streams many of its concerts live for free and keeps a video library of past ones, mostly song recitals, string quartets and solo piano. Lunchtime and evening streams are listed on its website. If you're staying in Marylebone or Mayfair it's also one of the easiest live evenings in London to fit round work, because the concerts are short, start at civilised times and have you home by ten.
Museums that open late
Weekday visitors lose the museums to office hours, but two big ones give you an evening back. The British Museum stays open until 8.30pm on Fridays, with last entry at 8.15pm. Most of the main galleries stay open, the Great Court and the Egyptian sculpture rooms included, though a handful close. It's free, but book a timed slot to be sure of getting in.
The V&A in South Kensington opens until 10pm on Fridays. On selected Fridays, usually the last of the month except in January, May and December, it runs Friday Late, a free drop-in evening of talks, music and installations built round a theme that pulls a younger crowd than the daytime museum. Get there before 8pm, when it fills up.
The London Marathon is in spring, usually late April, and the BBC shows it live. If you're near the route that weekend, watch the start on TV and walk out for the finish on The Mall, and check road closures before you book a car to the airport.
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These venues fall into three clusters: the South Bank for the National Theatre, Covent Garden for the Royal Ballet and Opera, and Marylebone for Wigmore Hall, with the British Museum a short walk north of Covent Garden in Bloomsbury. An apartment in Covent Garden, Fitzrovia or Marylebone puts at least two within walking distance and spares you the taxi queue at 10.30pm, when every show in the West End lets out at once. Nearly every central residence is within about ten minutes' walk of a station anyway, so the rest are rarely more than a short Tube ride off.
Before you settle in to watch
Apartment TVs vary. Ask before you arrive whether it's a smart TV with apps and whether you can cast to it from a phone or laptop, and if not, pack an HDMI cable, which weighs nothing and saves an evening of opera on a laptop screen. Ask about the Wi-Fi too. A shared building connection that copes fine with email can struggle with an HD stream at 9pm, and it's better to know before the curtain goes up.